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Properties

Label 2-432-108.11-c1-0-8
Degree $2$
Conductor $432$
Sign $0.843 + 0.537i$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.873 − 1.49i)3-s + (−0.268 + 0.738i)5-s + (3.49 + 0.616i)7-s + (−1.47 + 2.61i)9-s + (4.77 − 1.73i)11-s + (−4.34 + 3.64i)13-s + (1.33 − 0.243i)15-s + (6.45 − 3.72i)17-s + (−2.31 − 1.33i)19-s + (−2.13 − 5.77i)21-s + (−0.749 − 4.25i)23-s + (3.35 + 2.81i)25-s + (5.19 − 0.0767i)27-s + (3.09 − 3.68i)29-s + (2.76 − 0.487i)31-s + ⋯
L(s)  = 1  + (−0.504 − 0.863i)3-s + (−0.120 + 0.330i)5-s + (1.32 + 0.233i)7-s + (−0.491 + 0.870i)9-s + (1.43 − 0.523i)11-s + (−1.20 + 1.01i)13-s + (0.345 − 0.0627i)15-s + (1.56 − 0.904i)17-s + (−0.530 − 0.306i)19-s + (−0.465 − 1.25i)21-s + (−0.156 − 0.886i)23-s + (0.671 + 0.563i)25-s + (0.999 − 0.0147i)27-s + (0.574 − 0.685i)29-s + (0.496 − 0.0874i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.843 + 0.537i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.843 + 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31280 - 0.382491i\)
\(L(\frac12)\) \(\approx\) \(1.31280 - 0.382491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.873 + 1.49i)T \)
good5 \( 1 + (0.268 - 0.738i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-3.49 - 0.616i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-4.77 + 1.73i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (4.34 - 3.64i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-6.45 + 3.72i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.31 + 1.33i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.749 + 4.25i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.09 + 3.68i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.76 + 0.487i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.526 - 0.911i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.18 + 1.41i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.64 + 10.0i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.36 - 7.73i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 3.08iT - 53T^{2} \)
59 \( 1 + (5.15 + 1.87i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.62 - 9.24i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-6.27 - 7.47i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.72 - 2.99i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.99 - 8.64i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.929 + 1.10i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.12 + 0.946i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (3.10 + 1.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.87 - 1.04i)T + (74.3 - 62.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.54970987062246035190464477985, −10.36080052059303177619870598089, −9.121221483780744351227810552130, −8.193840520201160774482405655125, −7.25874017508678749613861636704, −6.54926098095846827694154310110, −5.34818644240852948321951090470, −4.40887737935071934960573681122, −2.57791829164237551129316698291, −1.24379815429714355952794892090, 1.35159598771173530713493780953, 3.46349188868424659325124147020, 4.58878927966449419709242953478, 5.19708463938513456564677069020, 6.39270228787531334578066241034, 7.73126998687959253723030956434, 8.482030232588913026158470243841, 9.683289045170305097367528251884, 10.26675655462792209423383656578, 11.21349420959409368483342419937

Graph of the $Z$-function along the critical line